Cannot make face, even with 4 vertices
You're not doing anything wrong, the tools you have chosen are working as expected, but perhaps they are not the right tools to achieve what you want. When you hit F, a face was created between the vertices around the back of your chair. If you use the little eye icon to switch off the Skin modifier, and switch on 'Limit Selection to Visible' you'll see it there, solid shaded. That face is the one that a modeler using F would usually be interested in. (i.e the faces between edges, not the ones a Skin modifier has put around edges)

The Skin modifier rapidly creates enclosing faces to a skeleton of edges and vertices, and with it, you can quickly convert those edges into an armature to control the deformation of the faces in animation. It tries its hardest to behave well at junctions, but things made with it are usually organic, and best kept simple.

To create a chair-frame, and learn some Blender at the same time, you could perhaps try a 'Box Modeling' approach .. I've just searched on 'Blender Box Modeling' and come up with some pretty reasonable looking tutorials.